How to get rid of this seam on my texture?
Things to check: 


Creased? Turn on the display of creases (N > Mesh Display > Creases), then enter Edit Mode (Tab) and look for a blue line along that section. If you see the blue line, clear the crease (Edge Select Mode, Ctrl+E, Clear Crease)
Internal Face? Check to see if you an internally extruded face along that section. Hit Z to enter Wireframe Mode, switch to Face Select Mode, and look for a third face attached to that edge. If so, delete it (them).
Edge Split? Ensure you don't have an Edge-Split modifier with too low of a value (Properties > Modifiers > Edge Split)
Shade Flat? Select all faces surrounding the issue and hit W then Shade Smooth




I downloaded your blend and took a quick look. It doesn't appear to be any of the things I've listed, but seem to be (as you suggested) associated with the UV mapped textures. The area in question is along two non-contiguous UV segments. So, unless the pixels directly under both segments have identical colors, one would absolutely expect there to be such a discontinuity.

My suggestion is to re-project the geometry so as to avoid seams in your UV in prominent positions.